Discussion of the Bob Schadewald Gravity Engine.

Bob's deception is so absurdly simple that everyone who has offered
public commentary on this has, so far as I know, missed it. The wheel does
gain kinetic energy on the downward half cycle; we can't deny that.
Bob then says that
the eccentric mass then has more than enough kinetic energy for it
to rise to the top again against the force due to gravity (which will
be slightly smaller while the mass rises).
That's also true. But does the wheel therefore
gain kinetic energy over the "long haul" of successive cycles, providing
excess energy we can tap?
Bob has cleverly led us to suppose it will. But if his logic and assumptions
are followed to their conclusion, we see that it will not.

Simple answer.

The only energy we could possibly extract from this system
would be that kinetic energy the ball
attains during its first fall to the floor, slightly less than
mgh.

More detailed analysis.

We will begin with an energy analysis, just to see what's going on. Suppose
that we do include some mechanism for extracting that extra amount of energy
at the end of each cycle. This returns the wheel to zero speed at the top, so
it begins from rest each cycle. But during each cycle the kinetic energy gain
at the top is
mh(gf - gi) = mhQ where
Q = gf - gi is the small change in
g during one cycle. Q is a negative
quantity, and is just equal to the change in
the potential energy at the top of the cycle due to the decrease of
g.

By the time n cycles are completed we have extracted
nQ = mgh of energy, the same amount we gave it initially
by lifting the weight to the top to get the wheel started. During this time
the speed at the bottom gradually decreases on subsequent cycles till it is
zero when gravity "runs out". We can get no more energy from this machine
than we put in initially.

There's a related puzzle in which we don't "steal energy" from the machine.
We also assume there are no dissipative forces (like friction) removing
energy from the machine.
Then when gravity reaches zero the final kinetic energy of the wheel must
be exactly what it was at the end of the first half-cycle.
Also, it must then be moving at constant speed at all positions.
That will be the same speed it had at the end of the first half-cycle.

Why did Bob say "the wheel may pick up speed at the top"? That will
happen if we don't extract any energy from the machine, or don't extract it
at a sufficient rate to ensure that the speed is zero at the top at the end
of each cycle.

Also, if g happens to reach zero when the eccentric mass is
anywhere but at the bottom, the mass will retain whatever speed it had
at that point, and
that includes the point at the top. This could be considerable if
g decreases so rapidly it reaches zero during the first few
cycles. But the speed at the top will never exceed the speed the mass had
when it first reached bottom.

It's interesting to note that in his Gravity Engine spoof, Bob Schadewald
never lies to the reader. He lets the reader make the incorrect
inferences. The closest he came to making an incorrect statement was "With
every revolution the wheel speeds up." Well, it does, at the top,
but never enough to cause its kinetic energy at any point
to exceed the kinetic energy it had attained when it first reached bottom.

We have used only conservation of energy (kinetic and potential) in this
discussion. The same result should be obtainable by a strictly kinematic
analysis using Newton's laws.

Related Puzzles:

How would steadily reducing gravity affect a simple pendulum?
Amplitude? Period? Velocity at bottom of swing? Height of swing?

How would steadily reducing gravity affect the motion of a mass suspended
from a Hookian spring (obeying Hooke's law), and bouncing up and down?